where \(\hat{\mu }_{-i}^{n+1}(\cdot )\)  is the estimated mean function based on a model fit to \(d_{-i}^{n+1}\) . Once the function \(\Psi \)  is specified by the data analyst, the same three-step IM construction in Section  can easily be followed to obtain a plausibility contour, \(\pi (y_{n+1};x_{n+1}, d^n)\) , whose computation is carried out analogous to that given in Algorithm REF ; see the box on page 388 of [1]}. A version of the key property (REF ) that drives both weak and strong prediction validity can be established in this case too,
\( \mathsf {P}\lbrace  \pi (Y_{n+1}; X_{n+1}, D^n) \le k_n(\alpha )\rbrace  \le \alpha , \quad \text{for all $(n,\alpha ,\mathsf {P})$}, \) 
